Contact geometry and quantum thermodynamics of nanoscale steady states

We develop a geometric formalism suited for describing the quantum thermodynamics of a certain class of nanoscale systems (whose density matrix is expressible in the McLennan–Zubarev form) at any arbitrary non-equilibrium steady state. It is shown that the non-equilibrium steady states are points on control parameter spaces which are in a sense generated by the steady state Massieu–Planck function. By suitably altering the system’s boundary conditions, it is possible to take the system from one steady state to another. We provide a contact Hamiltonian description of such transformations and show that moving along the geodesics of the friction tensor results in a minimum increase of the free entropy along the transformation. The control parameter space is shown to be equipped with a natural Riemannian metric that is compatible with the contact structure of the quantum thermodynamic phase space which when expressed in a local coordinate chart, coincides with the Schlögl metric. Finally, we show that this metric is conformally related to other thermodynamic Hessian metrics which might be written on control parameter spaces. This provides various alternate ways of computing the Schlögl metric which is known to be equivalent to the Fisher information matrix.